correct Set theory notations.

Question

What is correct notation for the following, I have seen both in some books.
To show an empty set, is it Φ(phi) or Ø(slash O) or both.
To show an Universal set, is it ε(epsilon) or U or both. I am little bit confused.

We use to denote the universal set, which is all of the items which can appear in any set. This is usually represented by the outside rectangle on the venn diagram.

http://www.cimt.plymouth.ac.uk/projects/mepres/book7/bk7i1/bk7_1i4.htm

Answer 1

The use of $\phi$ or $Ø$ is either from papers and books (and sometimes people) predating modern typesetting standards, or wrong (in term of standard notation). Or, of course, there is a confusion between the symbols because of the striking similarities between all of them.

The symbol $\varnothing$ and its variant $\emptyset$ are derived from the Norwegian-Danish letter Ø:

Common notations for the empty set include "{}", "Ø", and "\emptyset". The latter two symbols were introduced by the Bourbaki group (specifically André Weil) in 1939, inspired by the letter Ø in the Norwegian and Danish alphabet (and not related in any way to the Greek letter Φ).[1] Other notations for the empty set include "Λ" and "0".[2]

The empty-set symbol ∅ is found at Unicode point U+2205.[3] In TeX, it is coded as \emptyset or \varnothing.

(Wikipedia, Empty set)

As for universal sets, those are practically non-existent in modern set theory. I have seen them denoted by $U$ or sometimes $V$, but it might as well be any other letter.

The important thing is that the reader understand what each symbol means. Of course, if there are standard symbols it is better to use them. In the case of the empty set there is. In the case of a universal set, there isn't really a standard notation for it.